PROCEEDINGS
OF THE
AMERICAN MATHEMATICAL
Volume
103, Number
2, June
SOCIETY
1988
AN INEQUALITY FOR THE INTEGRAL
OF A HADAMARD PRODUCT
MEANS
MIROSLAV PAVLOVIÓ
(Communicated
ABSTRACT.
Motivated
Mq(r,f*g)
by Irwin Kra)
by Colzani's
paper [1] we prove that
< (l-r)l-^"\\f\\p\\g\\q,
0 < r < 1,
where 0 < p < 1, p < q < oo and / * g is the Hadamard
product
of / 6 Hp
and ge Ht.
For a function F, continuous
Mp(r,F)
in the disc U = {z : \z\ < 1} let
= ^j*\F(relt)\pdt,
0 < p < co,
Moo(r,P)=0maxjP(relt)|,
where 0 < r < 1. The Hardy class Hp consists of those / which are analytic
and satisfy the condition
in U
||/||p := sup Mp(r,f) < oo.
0<r<l
If f(z)
— J2anZn
and g(z) = J2bnzn
are analtyic
in U, then their Hadamard
product f * g, defined by
oo
(f *n)(z)
= }anbnzn
n=0
is analytic in U. It is well known that if f G H1 and g G Hq, q > 1, then
Mq(r,f * g) < H/Hillffllq(0 < r < 1) and consequently f * g G Hq. This fact is
generalized by the following theorem.
THEOREM. Let f GHP and g G H", where 0 < p < 1 and p < q < oo. Then
Mq(r,f*g)=0((l-r)1-1'p),
r-> 1~,
and,
(1)
Mí(r,/*9)<(l-r)1"1/"||/||P||9||0,
0<r<l.
For the proof we need a familiar lemma.
Received by the editors March 30, 1987.
1980 Mathematics Subject Classification (1985 Revision). Primary 30A10, 30D55.
Key words and phrases.
Integral
means,
Hadamard
product.
©1988
American
0002-9939/88
404
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INTEGRAL
MEANS OF A HADAMARD
405
PRODUCT
LEMMA. If F G Hp, 0 < p < 1, then
M^/H^l-rY-^IIFIIp,
0<r<l.
PROOF. Since
Mi(r,F)
< M^p(r,F)Mp(r,F),
we have to prove that
M0O(r,F)<(l-r2)-1/P||P||P.
This is reduced to the case p = 2 in the standard
way (see [2]). Let F(z) = J2cnZn
belong to H2. Then
Ml(r,F)< \T\en\A
OO
OO
0
0
and this concludes the proof.
PROOF OF THE THEOREM. It is easily shown that / and g may be supposed to
be analytic in the closed disc. By Parseval's formula
h(r2w) := (f * g)(r2w) = ±
[
f(re-ü)g(reltw)
dt,
\w\ = 1,
whence
\h(r2w)\ < -L j * \J^rT^)\\g(rétw)\dt
eilt
l- j
7r\F(reü)\dt
= Mi(r,F),
where
F(z) = f(z)g(zw),
\z\ < 1.
Since / is analytic in the closed unit disc we may apply the lemma to obtain
(i-sy-^w^^wFwi
= ^f*\f(e-ü)\p\g(ettw)\pdt.
Hence, by Minkowski's
inequality
(in continuous
form),
(1 - ra)1-»'Ma(ra, \h\p)
where s > 1. By taking s = q/p we get
(1 - r2y-pMp(r2,
and this concludes
h) = (1 - r2y-pMs(r2,
\h\p) < \\f\\pp\\g\\p,
the proof.
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MIROSLAV PAVLOVIÓ
406
REFERENCES
1. L. Colzani, Cesàro means of power series, Bull. Un. Mat. Ital. A (6) 3 (1984), 147-149.
2. P. L. Duren,
Theory of Hp spaces, Academic
Prirodno-Matematicki
Yugoslavia
Fakultet,
Press, New York, 1970.
Institut
za Matematiku
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nooo Beograd,